180 views

### Addition of two algebraic integer numbers [duplicate]

Is the addition of two algebraic integer numbers also algebraic? I, guess it is, but i can't prove it. I wonder if multiplication of them is also algebraic.
258 views

### Minimal polynomial for sum of algebraic numbers. [duplicate]

If I have two algebraic numbers $\alpha,\beta$ such that $A(\alpha) = 0$ and $B(\beta)=0$ where $A,B \in \mathbb{Q}[x]$ are the minimal polynomials of $\alpha$ and $\beta$ respectively. Knowing only ...
138 views

### If $\alpha$ and $\beta$ are roots of monic polynomials in $\mathbb{Z}[x]$, is $\alpha + \beta$? [duplicate]

If $\alpha$ and $\beta$ are roots of monic polynomials (not necessarily the same polynomial) in $\mathbb{Z}[x]$, is $\alpha + \beta$? I know that $\alpha + \beta$ will be the root of some monic ...
58 views

### Sum & Product of Two Algebraic Numbers [duplicate]

If you have algebraic numbers $x$ and $y$, and you know the polynomials of least degree of which each is a solution (written as a vectors of coefficients), then how is the vector of coefficients of ...
29 views

### Algebraic numbers theory [duplicate]

How I can prove that if there were $x$ and $y$ are algebraic numbers then $x+y$ and $x \cdot y$ are also algebraic numbers given that $x,y \in \mathbb R$?
8k views

### Enlightening proof that the algebraic numbers form a field

The proof I'm familiar with that the algebraic numbers $\mathbb A$ form a field uses the fact that the resultant of two polynomials $p,q\in\mathbb Q[x]$ satisfies the following properties: It is $0$ ...
1k views

### How to prove that algebraic numbers form a field?

I'd like to know how to prove algebraic numbers form a field, i.e., if $a,b$ are algebraic numbers, prove that $ab$ and $a+b$ are also algebraic numbers by finding specific polynomials that ...
769 views

### Decomposition of an algebraic number into a sum or product of algebraic numbers with smaller degree

An algebraic number can be identified by its minimal polynomial together with isolating intervals with rational bounds for its real and imaginary parts. The degree of an algebraic number is the degree ...
271 views

### How can I work with (i.e. add, multiply) algebraic numbers in practice?

I know that the real algebraic numbers $\mathbb A \subset \mathbb R$ form a field. I've seen this as a more theoretical result, but it's also seems nice idea to implement algebraic numbers for the ...
360 views

### a way to represent algebraic numbers in a computer

Say you want to represent the rational numbers in a computer. This is quite easy, you can think of them as pairs of integers. It is also easy to develop algorithms for adding, subtracting, multiplying ...
### Transcendence of $\sqrt{\pi}$
So it is known that $\pi$ is transcendental. With a little thought I was able to prove that $k\pi$ and $\pi^{k}$ for all $k\in\mathbb{Z}$ was transcendental. After that I thought about $\pi^{b}$ for ...